Numerically estimating acoustic transmission loss of a reactive muffler with and without mean flow

Accepted Manuscript Numerically Estimating Acoustic Transmission Loss of a Reactive Muffler with and without Mean Flow D.P. Jena, S.N. Panigrahi PII: DOI: Reference:

S0263-2241(17)30370-6 http://dx.doi.org/10.1016/j.measurement.2017.05.065 MEASUR 4791

To appear in:

Measurement

Received Date: Revised Date: Accepted Date:

10 June 2015 21 May 2017 26 May 2017

Please cite this article as: D.P. Jena, S.N. Panigrahi, Numerically Estimating Acoustic Transmission Loss of a Reactive Muffler with and without Mean Flow, Measurement (2017), doi: http://dx.doi.org/10.1016/j.measurement. 2017.05.065

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Numerically Estimating Acoustic Transmission Loss of a Reactive Muffler with and without Mean Flow D.P. Jena Department of Industrial Design, National Institute of Technology Rourkela, India-769008

S.N. Panigrahi School of Mechanical Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar-751 013, India

Abstract The present work numerically mimic the measurement techniques used to measure the acoustic transmission loss (TL) of a muffler without and with mean flow. First, the three-pole measurement method has been simulated in frequency domain for estimating acoustic TL of a muffler without mean flow. Along with advantages and disadvantages, the inability of such method in estimating acoustic TL in presence of mean flow is highlighted. Such restriction has been resolved by establishing the time domain simulation of three-pole and four-pole measurement methods. The requirement of anechoic termination for three-pole method is well known. Therefore, the issues involved in simulating such boundary condition numerically in frequency and time domain analyses have been addressed. The necessities of boundary layer meshing, selection of appropriate solver, impulse simulation, requirement of non-reflecting velocity inlet, simulation of two-load boundary condition, and role of microphone spacing have been discussed in the context of TL evaluation in the presence of mean flow. Additionally, some recommendations Email address: [email protected] (D.P. Jena)

Preprint submitted to Measurement

May 21, 2017

have been made to reduce the computational overhead in numerical analysis which is another key issue. Robustness of the proposed methods have been demonstrated by comparing estimated acoustic TL with experimental measurements. Keywords: Transmission Loss (TL), Three-Pole Method, Four-Pole Method, Transfer Matrix, Anechoic Termination, Non-reflecting Boundary Condition (NRBC), Mach number 1. Introduction Various experimental techniques to measure the acoustic transmission loss (TL) of a muffler without or with mean flow have been well established. However, they are highly time consuming and un-economical when the new mufflers are to be designed on a regular basis. Numerous analytical solutions have also been formulated by many researchers to estimate the acoustic TL with certain assumptions [1, 2]. However, such analytical solutions, though they are fast and economical, are heavily limited due to their inherent assumptions and severe geometry dependence. The desired compromise between these two extremities, from industrial point of view, is to estimate the acoustic TL using finite element software. The potential of the transfer matrix method has been exploited by many investigators in estimating acoustic TL of mufflers [1, 2, 3, 4]. Following such method, by multiplying transfer matrix of each sub-component of the muffler the resultant transfer matrix for the desired muffler can be estimated. However, from design perception, the evaluation of transfer matrix is comparatively difficult when the transfer matrix for a sub-component of the muffler is not known. Secondly, the empirical models have been used to model the acoustic impedances of perforated components are having certain assumptions. Summarizing, the estimation of acoustic TL using analytical solution or transfer matrix method is a very fast approach but result is not very close to experimental result in case of a complex muffler. 2

Considering aforementioned complexities, from industrial perspective, potential of available numerical software need to be exploited to analyze automotive mufflers without or with mean flow. One of such software namely R SYSNOISE⃝ , which is a boundary element (BEM) based software, has been widely used by many acoustic researchers to design the mufflers [2, 5, 6]. For such purpose, most of these investigators implement three-pole method in frequency domain analysis. However, the complexities involved in simulating acoustically anechoic termination, the pre-requisite of the three-pole method, in time domain analysis have not been reported in detail. Other R R available software such as ABAQUS⃝ , VNOISE⃝ are also capable of estimating acoustic TL of a muffler in frequency domain [7, 8]. A variety of numerical methods such as finite element method (FEM), boundary element method (BEM), finite volume method (FVM), finite difference method (FDM) and hybrid methods etc. also can be used to simulate the acoustics of a muffler and the corresponding advantages and disadvantages have been well acknowledged [5, 9, 10]. However, the numerical implementation of four-pole measurement method to eradicate the requirement of acoustically anechoic termination has not been addressed for various flow conditions [11]. A detailed comparison study between analytical and numerical methods in acoustic analysis of automotive silencers with mean flow has been reported by Kirby [12]. This report demonstrates the ability of numerical techniques in simulating acoustics of filters in presence of mean flow. In recent times, adjustable muffler analyses and micro-perforated partitioned mufflers have been analyzed [13, 14]. The higher order effect has also been discussed by many researchers, however, is not applicable in present analyses [15? ]. Summarizing, the issues involved in simulating acoustically anechoic termination in frequency and time domain analyses to mimic the three-pole measurement method needs to be addressed. From industrial prospective, a generalized stable numerical methodology should be established to estimate the acoustic TL of any arbitrary muffler for various flow condition. However,

3

this is not as straight forward as is claimed by many of these software. In the present work, above mentioned issues have been addressed in detail and a methodology to estimate the acoustic TL of a muffler has been recommended R using ANSYS⃝ . In the first section, frequency domain analysis has been carried out to evaluate the acoustic TL of a muffler using three-pole measurement technique. The complexities involved in simulating the anechoic termination have been addressed in detail. The limitations of such method in estimating acoustic TL in presence of mean flow has been pointed out and resolved by establishing time domain analyses of the three-pole and four-pole method using CFD. In subsequent section, the simulation of acoustically anechoic termination or non-reflecting outlet boundary condition has been investigated to establish the three-pole measurement method in time domain using density based solver. The necessity of anechoic termination boundary condition has been eradicated by establishing the four-pole method. The numerical instability observed using density based solver in achieving a steady state solution for various degrees of flow such as laminar and turbulent flow situation, has been addressed by adopting a pressure based solver in the last section. Using a pressure based solver, three situations such as (a) without flow, (b) flow with low Mach number, and (c) flow with high Mach number, have been investigated. 2. Simulation of Measurement Techniques The acoustic TL is defined as the ratio of incident and the transmitted sound wave with an assumption of anechoic termination. To estimate the acoustic TL under such an assumption, the three-pole measurement method has been proposed by Young and Crocker [16]. The schematic of such an arrangement has been shown in Fig.1(a). In the present work, a significant focus has been given to simulate the anechoic termination, which is a requirement of the three-pole measurement method. Thereafter, in order to eradicate the 4

requirement of anechoic termination, the four-pole measurement method has been adopted [11]. In the present case, the four-pole method using two-load condition has been demonstrated, which imitates the actual experiment. The schematic of such an arrangement has been shown in Fig.1(b). The higher order effect has been neglected, in the present analyses. 2.1. Three-Pole Method In three-pole method, two microphones are placed before the acoustic element and one microphone used to be placed after the element. It is assumed that no wave is reflected from outlet end. Under this assumption, the transmitted sound power, pt , can be easily obtained by measuring the sound pressure at the outlet. However, it is difficult to measure the incident sound power in presence of sound reflection from the acoustic element. The simulation of three-pole method can be executed in frequency domain by adopting harmonic analysis technique. In line with the practice, the TL can be estimated with a single numerical solution for a range of frequency with discrete frequency steps. The boundary conditions of a uniform sound pressure and characteristic acoustic impedance are required on inlet and outlet surface, respectively. From harmonic analysis, the extracted sound pressure at any point (x1 , x2 and x3 ) carries amplitude and phase (complex) information for all defined discrete frequencies. Mathematically, using the complex sound pressure level, P1 and P2 , from two points x1 and x2 , the incident pressure Pi can be calculated as [17]: ( ) Pi = P1 e−jkx2 − P2 e−jkx1 /2j sin{k(x2 − x1 )}

(1)

Using the incident pressure Pi and transmitted complex sound pressure P3 the TL can be computed using following equation: T L = 20 log10

|Pi | |Si | + 10 log10 |Pt = P3 | |So |

(2)

where Si and So are the cross-sectional areas of the inlet and outlet tubes 5

respectively. However, the time domain simulation (transient analysis) imitates the realtime experiment. Unlike previous simulation, the extracted sound pressure at any point (x1 , x2 and x3 ) carries only amplitude information on time axis, which are analogous to experimentally measured sound pressure signals. The one dimensional wave decomposition technique is incorporated to estimate the incident sound power [18]. The measured sound pressure signals, p1 and p2 , from position x1 and x2 (before the acoustic element), are decomposed into its incident and reflected spectra, Sa and Sb , respectively. The transmission loss (TL) in frequency domain can be computed as:  ( ) pi  T L = 20Log 10 pt  pi = √Sa ,and pt = √Sc

(3)

Here, the Sc is the auto spectrum of the transmitted sound pressure signal, p3 , measured at x3 position. Using the decomposition theory, the auto spectrum of the incident wave, Sa , can be evaluated as: Sa =

S11 + S22 − 2C12 cos(kx12 ) + 2Q12 sin (kx12 ) , 4 sin2 (kx12 )

(4)

where, S11 and S22 are the auto power spectra of the sound pressure signals at points x1 and x2 , respectively. C12 and Q12 are the real and imaginary parts of the cross power spectra of the sound pressure signals at points x1 and x2 . k is the wave number and x12 is the distance between the two microphones. In numerical simulation, the importance of the three-pole method is due to its low computational overhead. However, the complexity arises in simulating the anechoic termination or non-reflecting outlet boundary condition. This is a known problem in case of real-time experiment as well. 2.2. Four-Pole Method In order to get rid of the complexity involved in the three-pole method, the four-pole method is widely used to measure acoustic TL [1, 11] of the muf6

flers. Sound pressure signals from four predefined points are used to evaluate the transfer matrix coefficients as shown in Fig.1(b). Four-pole method can be implemented either with the two-load or the two-source condition. In the present work, the simulation has been carried out using the two-load boundary condition. In case of actual experimental measurement, the pressure p and particle velocity u with two different load conditions, ‘a and ‘b’ (refer Fig.1(b)) are used to solve the linear equations to obtain the four unknown matrix elements [1], where ′ 0′ and ′ d′ stands for m2 and m3 microphone location. [

p0a u0a

[

p0b u0b

]

[ =

x=0,Load=a

]

[ =

x=0,Load=b

T11 T12 T21 T22 T11 T12 T21 T22

][

][

pda uda pdb udb

] (5) ]

x=d,Load=a

(6) x=d,Load=b

First, the sound pressure p and particle velocity u for load condition ‘a’ are evaluated using the sound pressure signals from four points. The first microphone is considered as the reference microphone and the corresponding transfer function H11 is 1. Transfer functions of other three microphones with respect to the first microphone can be evaluated as mentioned below. H21 =

C12 , S22

C13 ,and S33

H31 =

H41 =

C14 S44

(7)

where C1N is the cross power spectrum between reference microphone with other microphone locations and SN N is the auto power spectrum of the sound pressure at other microphone locations (N = 2, 3, 4). For each load case, the acoustic wave field inside the tube can be decomposed as forward and backward traveling waves using the below mentioned equations [11, 1]. {

A = j H11 e B=

−jkl1 −H e−jk(l1 +s1 ) 21

, C = j H31 e

2 sin(ks1 ) H21 e+jk(l1 +s1 ) −H11 e+jkl1 j , 2 sin(ks1 )

7

D=

+jk(l2 +s2 ) −H e+jkl2 41

2 sin(ks2 ) H41 e−jkl2 −H31 e−jk(l2 +s2 ) j 2 sin(ks2 )

(8)

A and B are the complex amplitudes of the forward and backward traveling waves in the tube upstream of the acoustic element. Similarly, C and D are those in the tube downstream of the acoustic element. The acoustic pressure and particle velocity upstream and downstream of the acoustic element may be computed as mentioned below. {

p0a = A + B, pda = Ce−jkd + De+jkd −jkd −De+jkd u0a = A−B , uda = Ce ρc ρc

(9)

In a similar way the upstream and downstream acoustic pressure and particle velocity (p0b , u0b , pdb , udb ) for load condition b are evaluated. Now, using the acoustic pressure and particle velocity for both the load cases, the transfer matrix for the acoustic filter can be computed as: [ T =

p0a udb −p0b uda pda udb −pdb uda u0a udb −u0b uda pda udb −pdb uda

p0b pda −p0a pdb pda udb −pdb uda pda uob −pdb u0a pda udb −pdb uda

] (10)

Using the transfer matrix coefficients, the acoustic transmission loss of the acoustic filter can be evaluated as:   T L = 20Log10 1 t 2ejkd  t= T12 T11 +

ρc

(11)

+ρcT21 +T22

In the present exercise, a simulation has been carried out to exactly imitate the experimental measurement method. The time domain sound pressure signals from four points, which corresponds to four microphones in the experiment, under two different outlet boundary conditions have been used to estimate the acoustic TL. 3. Frequency Domain Simulation 3.1. Numerical model for acoustic analysis Acoustic analysis, in our context, is based on the postulation such as an enclosed volume of fluid, which is compressible, in-viscous, without mean flow 8

and with uniform mean density and pressure. Along with this, the enclosure walls are assumed to be ideally rigid. Based on the above assumptions, the acoustic wave equation may be written as: 1 ∂ 2p , (12) c2 ∂t2 where, c is the speed of sound in the medium, p is acoustic pressure, t is time and ∇2 is Laplace operator [1]. Since the viscous dissipation is neglected, the above expression is also considered as loss-less wave equation for sound propagation. Then, for a small change in pressure, δp, over a finite fluid volume, the wave equation can be represented in the integral form as: ∇2 p =

∫ V

1 ∂ 2p δp dV + c2 ∂t2

∫

∫ ({L} δp)({L}p)dV −

{n}T δp({L}p)dS = {0}, (13)

T

V

S

where V is the volume of the fluid domain, S is the surface where the derivative of pressure normal to the surface is applied and {n} is the unit vector normal to the surface S. In simulating fluid-structure interaction, the surface S is treated as the fluidsolid interface. The normal pressure gradient of the fluid and the normal acceleration of the structure at the fluid-structure interface S can be expressed as: ∂ 2 {u} {n}.{∇p} = −ρ0 {n}. , (14) ∂t2 where {u} is the displacement vector of the structure at the interface. Now, the Eq.13 can be rewritten as: ) ∂2 ({L} )({L}p)dV + ρ0 δp{n} {u} dS = {0} ∂t2 V V S (15) In finite element modeling, using element shape function, {N }, for pressure, ∫

1 ∂ 2p δp dV + c2 ∂t2

∫

(

∫

T

T

9

element shape function, {N ′ }T , for displacements, and nodal pressure vector {¨ pe }, the second order derivatives can be represented as: ∂2p = {N }T {¨ pe } ∂u2

(16)

∂2 {u} = {N ′ }T {¨ ue } ∂u2

(17)

where, p = {N }T {pe } and u = {N ′ }T {ue }. Now, the discretized wave equation can be written in matrix notation as: [MF ]{¨ pe } + [KF ]{pe } + ρ0 [R]T {¨ uF,e } = {0}, where, 1 [MF ] = 2 c

(19)

[∇N ]T [∇N ]dV

(20)

{N }{n}T {N ′ }T dS

(21)

[KF ] =

[R]T =

∫ {N }{N }T dV

∫ ∫

(18)

V

V

S

The [MF ], [KF ], [R]T are acoustic fluid mass matrix, acoustic fluid stiffness matrix and acoustic fluid coupling matrix, respectively. Assuming time harmonic input, p = pejωt , the Eq.12 can be re-written as: ∇2 p + k 2 p = 0,

(22)

where, k = (ω/c) is wave number, ω is the angular frequency (ω = 2πf /c). 3.2. Estimating Acoustic TL To establish the process of evaluating TL, a benchmark muffler configuration has been considered for analysis using ANSYS Parametric Design Language (APDL). The 3D model of the simple expansion chamber and the corre10

sponding FEM model have been shown in Figs.2(a) and 2(b), respectively. In the present investigation, the harmonic analysis has been carried out from 10 Hz to 3 KHz with step size of 20 Hz. The FLUID-30 element, a eight node element, has been used for acoustic analysis. As discussed earlier, the mathematical model neglects the fluid flow field. So the boundary layer meshing is not required and recommended. The acoustic element has three input parameters such as the density of the material, sound speed in the material, and the boundary admittance at the interface. The density of the air and the speed of the sound in air have been defined as 1.2 kg/m3 (at 20o C) and 343 m/s, respectively. The walls have been considered as highly reflective, so the boundary admittance has been neglected. The sound pressure of 10 Pa has been applied on inlet surface as nodal load to simulate the input sound pressure level. The issue involved in simulating anechoic termination has been addressed by applying characteristic acoustic impedance on outlet boundary. However, the numerical error may be observed if the boundary loads are not been applied on surface nodes (load on a plane boundary) as shown in Figs.2(c) and 2(d). Now, in outlet surface, the characteristic acoustic impedance (rhoair ∗cair ) have been applied to simulate the acoustically anechoic termination. The sound pressure (complex signals) from nodal solution at three microphone positions (positions x1 , x2 and x3 ), as shown in Fig.3 (absolute signals), have been used in Eq.(1) and Eq.(2) to evaluate the acoustic TL of the muffler and has been compared with experimental measurements, using B&K transmission loss tube, shown in Fig.4. The strength of the proposed method is that any complicated acoustic filter can be analyzed by only modeling acoustic cavity with the minimum element quality such as the maximum desired frequency wavelength should be 5-8 times of the element length. However, the proposed methodology has the limitation in estimating acoustic TL in presence of mean flow.

11

4. Time Domain Simulation The necessity of time domain analysis has been observed to evaluate the acoustic TL in presence of mean flow. In the present work, the time doR main analysis has been investigated using FLUENT ⃝ CFD solver. In all flow situations, the CFD solver solves the conservation equations for mass and momentum. As the sound wave propagates due to the compressibility nature of the fluid, an additional equation such as conservation of energy is being solved. In the present study, an ideal gas model has been adapted for simulating the sound propagation in air. In order to reduce the computational overhead a 2D-axisymmetric geometry of a simple expansion chamber has been investigated. From the literature, one can find that the density based solver is appropriate to simulate the propagation of compressible fluid in the absence of mean flow. This assumption is highly desired to satisfy the Helmholtz wave equation [19, 20, 21]. The subsequent sections explain the mathematical assumptions and the solver formulations to achieve acoustic wave propagation in the fluid (ideal gas). 4.1. Governing equations The generalized form for conservation of mass for 2D-axisymmetric geometries can be written as [22, 19]: ∂ρ ∂ ∂ ρvr + (ρvx ) + (ρvr ) + = Sm (23) ∂t ∂x ∂r r For 2D-axisymmetric geometries, the axial and radial momentum conservation equations can be written as:

∂ ∂t

∂ ∂ (ρvx ) + 1r ∂x (rρvx vx ) + 1r ∂r (rρvr vx ) [ ( )] ∂P 1 ∂ ∂vx 2 = − ∂x + r ∂x rµ 2 ∂x − 3 (∇ · ⃗v ) +

12

1 ∂ r ∂r

[ ( ∂vx rµ ∂x +

∂vr ∂x

)]

+ Fx

(24)

∂ ∂t

∂ ∂ (ρvr ) + 1r ∂x (rρvx vr ) + 1r ∂r (rρvr vr ) [ ( )] 1 ∂ [ ( ∂vr 2 )] ∂P 1 ∂ ∂vr ∂vx = − ∂r + r ∂x rµ ∂x − ∂r + r ∂r rµ 2 ∂r − 3 (∇ · ⃗v )

−2µ vr2r +

2µ 3r

(∇ · ⃗v ) + ρ

2 vZ

r

(25)

+ Fr

where, ∂vx ∂vr vr + + (26) ∂x ∂r r Compressible fluid flow using an ideal gas satisfies the physics of acoustic wave propagation. The energy equation for compressible fluid flow (ideal gas) can be formulated as: ∇ · ⃗v =

∂ (ρE) + ∇. (⃗ν (ρE + p)) = ∇. (k∇T + τ¯.⃗ν ) ∂t ∑ p ν2 E= Yj hj − + ρ 2 j

(27) (28)

4.2. Numerical model for acoustic analysis: Density Based Solver The density based solver has been well established to simulate the compressible fluid model. The density-based solver solves the governing equations, mentioned earlier, such as continuity, momentum, and energy equations, simultaneously. In Cartesian system, for a single phase compressible fluid model, the vector integral form for an arbitrary control volume (V ) with differential surface (dA) can be formulated as [23]: ∂ ∂t

∫ ∫ ∫

∫ ∫ W dV +

∫ ∫ ∫ [F − G] .dA =

HdV

(29)

where the vectors W , F and G are characterized as:      ρ  ρν             ˆ    ρu    ρνu + pi W = ,F = ρνu + pˆj ρν         ρw  ρνu + pkˆ             ρE ρνu + pν 13

              

        ,G =

      

0 τxi τyi τzi τij νj + q

              

(30)

Here the ρ, ν, E, and p are density, velocity, total energy per unit mass, and pressure of the fluid, respectively. The τ is the viscous stress tensor, and q is the heat flux. The total energy E and total enthalpy H are related as: {

E=H− H =h+

p ρ |ν|2 2

(31)

The numerical stiffness has been observed in solving the finite volume model of Navier-Stokes equations. The inconsistency between fluid velocity (ν) and acoustic speed c results such stiffness. To overcome such situation, the time-derivative based preconditioning has been employed in above mentioned density based solver. In such technique, an additional term has been multiplied. Using chain rule the dependent variables have been transformed such as conserved quantities W to primitive variables Q as: ∂W ∂t

∫ ∫ ∫

∫ ∫ QdV +

∫ ∫ ∫ [F − G].dA =

HdV

(32)

where Q is the vector [p, u, ν, w, T ]T and the Jacobian term ∂W /∂Q can be expressed as:     ∂W =  ∂Q  

where,

ρp 0 0 0 ρT ρp u ρ 0 0 ρT u ρp ν 0 ρ 0 ρT ν ρp w 0 0 ρ ρT w ρp H − 1 ρu ρν ρw ρT H + ρCp   ρp =  ρT =

       

(33)



∂ρ ∂p T ∂ρ ∂T p

(34)

To satisfy, the Helmholtz wave propagation assumption, pressure inside the compressible fluid flow should be a dependent variable.

14

4.3. Simulation of Anechoic Termination In order to achieve the anechoic termination, the elimination of reflection from outlet boundary is desired. The general non-reflecting boundary conditions (NRBC) have been computed from the Euler equations using characteristic wave relations. In the density-based solver, the non-reflecting boundary conditions can be applied only on pressure-outlet boundary using the ideal gas model [24]. The modified Euler equations have been solved on boundary to extract the primitive flow quantities (p, u, ν, w, T ). The modified Euler equations for the non-reflecting boundary can be written as:                     

∂ρ 2 3 + d1 + ∂m + ∂m =0 ∂t ∂x2 ∂x3 ∂m1 1 U2 1 U3 + U1 d1 + ρd3 + ∂m + ∂m =0 ∂t ∂x2 ∂x3 ∂p ∂m2 ∂m2 U2 ∂m2 U3 + U2 d1 + ρd4 + ∂x2 + ∂x3 + ∂x ∂t 2 ∂p ∂m3 ∂m3 U2 ∂m3 U3 + U d + ρd + + + 3 1 5 ∂x2 ∂x3 ∂x3 {∂t ∂ρE d2 1 2 + |V | d + + m d + m 1 1 3 2 d4 ∂t 2 (γ−1) ∂[(ρE+p)U2 ] 3] + ∂[(ρE+p)U =0 ∂x2 ∂x3

=0 =0 + m3 d5 +

(35)

Where m1 = ρU1 , m2 = ρU2 and m3 = ρU3 . The U1 , U2 and U3 are the velocity components of the corresponding Cartesian coordinate system (x1 , x2 , x3 ). The above mentioned equations have been solved for non-reflecting boundary. The solver solves parallel the interior governing flow equations as mentioned earlier to obtain the primitive flow variables values (p, u, ν, w, T ). The di terms in the transformed Euler equations contain the outgoing and incoming characteristic wave amplitudes Li . The terms di and Li are defined as [25]:  ( 1 d = L2 +  1 2  c   L +L 5 1    d2 = 2 −L1 d3 = L52ρc     d4 = L3    d5 = L4

15

L5 +L1 2

)

(36)

 ( ) ∂p ∂U1  L = λ − ρc  1 1 ∂x1 ∂x1   ( )   ∂ρ ∂p 2    L2 = λ2 c ∂x1 − ∂x1 2 L3 = λ3 ∂U ∂x1    3  L4 = λ4 ∂U  ∂x 1  ( )    L5 = λ5 ∂p + ρc ∂U1 ∂x1 ∂x1

(37)

The outgoing and incoming characteristic waves are associated with the characteristic velocities of the domain, i.e., eigen values, λi . These eigen values are computed by:    λ1 = U1 − c λ2 = λ3 = λ4 = U1   λ5 = U1 + c

(38)

First, the above mentioned equation (25) has been solved to evaluate the amplitude of the incoming and outgoing waves. The extrapolated flow derivatives ∂p/∂x1 , ∂ρ/∂x1 , ∂U1 /∂x1 , ∂U2 /∂x1 , and ∂U3 /∂x1 for inside domain. Next, the amplitudes of outgoing waves (non-reflecting boundary) have been computed using equation (26) and the evaluated extrapolated flow derivatives. The tangential velocity components such as amplitude of waves L3 , and L4 have been set to zero. The amplitude of the incoming pressure wave and the corresponding entropy have been derived using the Linear Relaxation Method (LRM) [26]. In general, the required pressure of a non-reflecting boundary can be either relaxed to a pressure value at infinity (P∞ ) or forced to be equivalent with average pressure of the exit boundary. In present case, the average pressure of the non-reflecting boundary (pressure outlet) has been relaxed infinity, i.e., pexit = p∞ .

16

4.4. Simulation of a Progressive Wave in a Duct with NRBC outlet In order to achieve anechoic termination, the simulation of non-reflecting R outlet boundary is foremost desired. The inadequacy of FLUENT⃝ in simulating the non-reflecting boundary condition has already been reported by Torregrosa et al. in his recent report [27]. In order to establish the process of imposing a non-reflecting outlet boundary, simulation of a progressive plane wave inside a duct has been carried out. The 2D-axisymmetry geometry of the duct and the corresponding mapped mesh along with boundary-layer mesh have been shown in Fig.5. The inlet and the outlet of the duct have been assigned as pressure inlet and pressure outlet boundary conditions, respectively. The non-reflecting boundary condition has been applied on the outlet boundary with the “pressure to infinity” option while specifying exit pressure. The analysis has been carried out using the density based solver with explicit time stepping scheme. After the standard initialization, a pressure fluctuation of 500 Hz has been applied on the inlet boundary using a user defined function (UDF). In transient analysis, the desired time-step of the simulation is 2e-4 seconds (for mapped mesh), following below mathematical expression [19].   ∆tP =  U∆P =

0.3×Lscale

√U(∆P

Pbc,max −Pbc,min ρ¯cells

)

(39)

Here, Pbc,max and Pbc,min are the maximum and minimum pressure values at open boundary, respectively, and ρ¯ is the average density over the domain. The effect of choosing the adequate time-step has been demonstrated by using different time steps, such as 2e-4, 1e-4, 1e-5 and 1e-6 seconds, respectively, for mapped mesh and boundary-layer mesh files. Numerical instability has been observed on using the time step size of 2e-4 or 1e-4 seconds. However, the results converged on using the smaller time steps. The observed pressure fluctuations in the duct for different chosen time steps, using mapped and boundary layer mesh, have been shown in Figs.6(a-d), respectively. Similarly, 17

the observed pressure profile along the axis have been shown in Figs.7(a,b), respectively. From the Fig.6, it is worth noticing that the progressive plane wave propagates adequately using boundary layer meshing and mapped mesh for time step of 1e-6 seconds. However, at higher time step of 1e-5 second, the numerical error has been observed for boundary layer meshing. This could be due to the presence of smaller elements near the boundary. In the present analysis, there is no mean flow and the plane wave propagation is the only requirement to estimate the acoustic TL. In order to reduce the computational overhead, the mapped mesh is recommended. Simulation of non-reflecting boundary condition is highly dependent on the time step size (see Figs.6 and 7). Higher numerical accuracy can be achieved by using smaller time steps. However, the selection of an appropriate time step to compromise the computational overhead is to be again decided by the investigator. In the present exercise, all subsequent time domain analyses have been carried out using the time step size of 1e-6 second. 5. Simulation without Mean Flow From the above analysis, it is evident that the simulation of anechoic termination is achievable, at least, in absence of flow by imposing the non-reflecting boundary condition on the pressure outlet boundary and choosing an appropriate time step. Next, the challenge of estimating the acoustic TL of a filter using three-pole method has been taken up. The geometry and the corresponding mapped mesh have been shown in Figs.8(a,b), respectively. The inlet and the outlet of the muffler have been modeled with longer ducts to allow the development of plane wave from a pulse input. The inlet and the outlet of the muffler have been assigned as pressure inlet and pressure outlet boundary conditions. The non-reflecting boundary condition has been applied on the outlet boundary with the “pressure to infinity” option as discussed earlier. Using the density based solver along 18

with the explicit time stepping of 1e-6 seconds, a transient analysis has been carried out. Initially, a few iterations have been executed with gauge pressure applied on the inlet. Next, a pressure fluctuation (a rectangular pulse) has been applied on the pressure inlet. After the pressure fluctuation, the inlet has again been assigned to gauge pressure and the transient analysis has been carried out for 0.05 seconds of real-time. In order to estimate the acoustic TL, the pressure signals from four points (see Fig.1(b)) have been captured during the transient analysis. As mentioned earlier, for the three-pole measurement method, only three pressure signals (see Fig.1(a)) need to be used and the same have been shown in Figs.9(a-c), respectively. The estimated acoustic TL following the three-pole method has been shown in Fig.10. From Fig.10, it is worth noticing that the estimated acoustic TL agrees to the experimental results adequately. In order to eradicate the requirement of anechoic termination (as a preparation for the analysis with mean flow), next, the four-pole measurement method has been used with two-load boundary conditions. For estimating the acoustic TL using the four-pole method, the four pressure signals (see Fig.1(b)) have been used. As discussed earlier, to simulate two loads, the transient analyses have been carried out with an open-end and then with a closed-end boundary condition. The open outlet boundary condition has been simulated by defining the outlet as a simple pressure outlet, and the closed boundary condition has been simulated by defining the outlet boundary as a wall. The corresponding pressure signals in time domain have been shown in Figs.11 and 12, respectively. The four corresponding pressure signals for a non-reflecting boundary have already been shown in Fig.9. The acoustic TL using four-pole with two-load method has been computed. As a first attempt, the two types of load conditions considered are nonreflecting and the open-end boundaries. Next, the two types of load conditions have been changed to the non-reflecting and the closed-end boundaries. Lastly, the open-end and the closed-end boundaries have been considered to

19

be the pair of load condition. The acoustic TL curves evaluated from all above three conditions have been shown in Fig.13. It is worth noticing that all the estimated acoustic TL curves agree to experimental result adequately. The four-pole measurement method extracts the sound pressure ‘p’ and particle velocity ‘u’ upstream and downstream of the acoustic filter to estimate the acoustic TL. In order to reduce the computational overhead, the numerically extracted pressure and velocity can directly be used in Eq.10 to get the transfer matrix. Towards this, the axial particle velocities on positions m1 and m3 (see Fig.1(b)) for different outlet conditions have been acquired and shown in Fig.14. Subsequently, only the sound pressure at positions m1 and m3 together with the corresponding axial particle velocities have been used to form the transfer matrix. The estimated acoustic TLs have been shown in Fig.15. From the above analysis, the estimation of acoustic TL using three-pole technique with an assumption of anechoic termination on outlet has been established. However, the observed numerical errors can be eliminated by adopting the four-pole technique. The above analyses also demonstrate that the requirement of anechoic termination (NRBC outlet) can be eliminated by implementing the four-pole with two-load (with open-end and closed-end load conditions) measurement method. However, some fluctuation in estimated acoustic TL has been observed at low frequencies (≤ 500 Hz). Still the challenge persists in eliminating the requirement of non-reflecting boundary condition. Because, the closed boundary condition cannot be applied in presence of mean flow. The appropriate two-load boundary conditions in presence of mean flow have been discussed in the next section. 6. Simulation with Mean Flow Next level of complexity arises, when a flow is present across the filter. The earlier methodology has the limitation of estimating the acoustic TL only in the absence of any mean flow. Once, the mean flow has been taken in account, 20

several issues come into sight. For example, the density based solver does not work efficiently [28, 29, 20] to simulate the fluid flow. Moreover, in order to establish the three-pole method, simulating the assumption of anechoic termination on a mean flow outlet is numerically unstable. In addition, the four-pole−two-load method cannot be established with the open-and closedend outlet load condition pair. In order to evaluate the acoustic TL of a filter with flow, first the pressure based solver has been used and validated as explained next. 6.1. Numerical model for acoustic analysis: Pressure Based Solver The traditional practices associated to discretization of the momentum and continuity equations and evaluating their solutions by using the pressurebased solver are discussed. Mathematically, The steady-state continuity and momentum equations in integral form [28, 20] can be expressed as:  H H H H ⃗ = − pI · dA ⃗ + τ¯ ·¯dA ⃗ + F⃗ dV ,  ρ⃗v⃗v · dA V  H ρ⃗v · dA ⃗=0

(40)

where I is the identity matrix, τ¯ is the stress tensor, and F⃗ is the force vector. The discretization scheme used to discretize the scalar transport equation such as momentum equation for x-axis can be written as: aP u =

∑

anb unb +

∑

pf A.ˆi + S

(41)

nb

The velocity field can be obtained by solving above equations with known pressure field and face mass fluxes. However, the pressure field and mass fluxes for a face are not known and must be obtained as a part of the numerical solution. For that reason, an interpolation scheme is required to compute the pressure of face from the cell values.

21

6.2. Performance of Pressure Based Solver in Estimating Acoustic TL without Mean Flow In order to evaluate the performance of the pressure based solver, estimation of the acoustic TL without any mean flow is carried out as a special case. The methodology is quite similar to the one discussed earlier for the density based solver with some minor variations. In an attempt to come up with a generalized flow independent boundary condition, unlike the earlier analysis, the inlet of the filter has been defined as a velocity inlet and the outlet has been defined as a pressure outlet. From literature, one can find various suggestions on assigning the additional non-reflecting boundary condition on velocity inlet to simulate the plane wave [26, 30]. In line with that, velocity inlet also has been modeled to have the non-reflecting velocity property. Next, a velocity fluctuation (rectangular pulse) has been applied on velocity inlet. After applying fluctuation, the velocity inlet has been again assigned to initial velocity (present case it is zero) and apply non-reflecting boundary condition. The pressure signals from four points have been captured during the transient analysis and have been shown in Fig.16. The estimated acoustic TL following the three-pole method has been shown in Fig.17. It is worth noticing that the estimated acoustic TL agrees reasonably to the experimental results. Next, the four-pole with two-load method has been carried out and the acoustic TL has been evaluated. The estimated acoustic TL curves have been shown in Fig.18. That establishes the process of estimating acoustic TL using the three-pole, and the four-pole methods using the pressure based solver when no mean flow is present. When it comes to analyzing filter performances in presence of the mean flow, the closed-end boundary condition is not applicable. Therefore, the four-pole−two-load method has been simulated by attaching another filter on the outlet. However, the additional filter can be the same filter as the one which is under investigation. Next, a filter attached with

22

another filter on the outlet has been taken for analysis and has been shown in Fig.19(a). The other analysis can be done with the pressure outlet boundary condition. The acquired signal at the four microphone locations have been shown in Fig.19(b-e). The estimated acoustic TL has been shown in Fig.20. From the figure, it can be observed that the estimated acoustic TL agrees with the experimental result. From the above analysis, it is evident that one type of load condition for four-pole with two-load measurement method can be simulated with a nonreflecting velocity inlet and open-end pressure outlet condition. By adopting another filter on the outlet, the other load condition can be simulated. This methodology allows us to simulate the acoustic filter with mean flow. 6.3. Estimating Acoustic TL with Mean Flow: Low Mach Number Subsequently, a simulation has been carried out to estimate acoustic TL with presence of mean flow. In first level, the analysis has been investigated with very low Mach number such as 2.92e-5. In order to reduce the computational overhead, using the steady state solution, a transient analysis has been initialized and computed. In line with proposed methodology, first, the steady state solution for a given mean flow is desired. It is well established that the boundary layer meshing is desired to simulate the flow problem. Hence a boundary layer mesh file has been generated for analysis as shown in Fig.21. First, the mean flow velocity has been applied on the velocity inlet. The out let has been simulated as pressure outlet. In the present analysis, the pressure-based solver solves the fluid flow field in segregated manner using SIMPLE algorithm, based on the predictor-corrector approach. The default configuration for spatial discretization methods, and solution controls have been used to solve the steady state solution. A hybrid initialization technique has been used in order to achieve faster solution followed by executing 5000 number of iterations to reach the steady state solution. The pressure and velocity contour of filter without any outlet impedance and filter with additional impedance have been 23

shown in Fig.22, respectively. The steady solutions have been exported to data file and been stored in local memory. Now, the transient analysis has been initiated with certain steps of initialization. First, the velocity inlet has been assigned to given velocity (used in steady state solution) followed by hybrid initialization. Next, the initialization has been carried out by using the data file stored from steady state solution. Subsequently, the transient analysis has been executed with auto time step decided by solver. Next, the initialized solution has been analyzed for few rounds to achieve a stable solution. This initial analysis can be done with bit higher time steps followed by desired time steps. In the present analysis, after initializing from steady state solution, the transient analysis has been carried out for 1000 steps with 1e-4 seconds of time step. Next, similar to earlier analysis, a velocity fluctuation has been applied on velocity input (rectangular pulse). After applying the velocity pulse, the velocity inlet has been assigned to given velocity (used in steady state solution) and assigned to non-reflecting velocity inlet. The non-reflecting velocity inlet boundary condition helps in simulating the acoustic reflection from the acoustic filter adequately. Next important factor is position of four-poles (microphone positions) for capturing sound pressure signals. The microphone positions depend upon the desired frequency range and the flow velocity following below relation [31]. ( ) ( ) 0.1π 1 − M 2 < ks < 0.8π 1 − M 2

(42)

Where M is the Mach number, s is the microphone spacing and k is the wave number. The simulated four-pole signals for acoustic filter without outlet impedance have been shown in Fig.23 and for acoustic filter as outlet impedance have been shown in Fig.24. The estimated acoustic TL following four-pole method has been shown in Fig.25. From Fig.25, it is worth noticing that the estimated acoustic TL agrees to experimental result (without flow) 24

adequately due to flow at significantly low Mach number. 6.4. Estimating Acoustic TL with Mean Flow: High Mach Number Now, the most challenging issue is to estimate the acoustic TL when the flow is turbulent or at high Mach number. In general, the industrial automotive filters have been designed to perform with Mach number from 0.05 to 1.5. Next, a filter has been analyzed with a velocity input of 20.58 meter/second (Mach number of 0.06). Similar to earlier methodology, first, the steady state solution is desired. The standard k − ϵ model has been used to simulate the turbulent flow. The two-equation based turbulence model allows the determination of both the turbulent length and time scale by solving two separate transport equations. In the derivation of the k − ϵ model, the assumption is that the flow is fully turbulent, and the effects of molecular viscosity are negligible. The standard k − ϵ model is based on the transport equations for the turbulence kinetic energy (k) and its dissipation rate (ϵ). The model transport equation for k is derived from the exact equation, while the model transport equation for ϵ was obtained using physical reasoning and bears little resemblance to its mathematically exact counterpart. Mathematically, the turbulence kinetic energy, k, and its rate of dissipation, ϵ, are obtained from the following transport equations [32]:

∂ ∂ ∂ (ρk) + (ρkui ) = ∂t ∂xi ∂xj

[(

µt µ+ σε

)

)

] ∂k + Gk + Gb − ρε − YM ∂xj

(43)

] ∂ε ε ε2 + C1ε (Gk + C3ε Gb ) − C2ε ∂xj k k (44) In these equations, Gk represents the generation of turbulence kinetic energy due to the mean velocity gradients and Gb is the generation of turbulence kinetic energy due to buoyancy. YM represents the contribution of the fluctu∂ ∂ ∂ (ρε) + (ρεui ) = ∂t ∂xi ∂xj

[(

µt µ+ σk

25

ating dilatation in compressible turbulence to the overall dissipation rate. C1ϵ , C2ϵ , and C3ϵ are constants. σk and σε are the turbulent Prandtl numbers for k and ϵ, respectively. The turbulent (or eddy) viscosity, µt , is computed 2 by using relation µt = ρCµ kε , where Cµ is a constant. The default values for model constants are 1.44, 1.92, 0.09, 1 and 1.3 for C1ϵ , C2ϵ , C3ϵ , σk and σε , respectively. In order to demonstrate the estimation of acoustic TL, a filter configuration has been taken from literature [31]. The central chamber length and diameter of the simple expansion chamber are 0.5m and 0.2m, respectively. The inlet and outlet are identical of diameter 0.067m and lie on the center of the axis. Similar to earlier analysis, the corresponding boundary layer mesh file has been analyzed. The mean flow velocity such as 20.58 (Mach No. 0.06) has been applied on the velocity inlet. The out let has been simulated as pressure outlet. In present analysis, the pressure-based solver solves the flow problem in segregated manner using coupled pressure-velocity algorithm. The default spatial discretization methods and solution controls have been used to solve the steady state solution. However, the steady state solution can be achieved by adapting higher CFL number. In present case, the CFL number of 200 is observed numerically stable and has been used. A hybrid initialization technique has been adapted in order to achieve faster solution followed by executing 5000 number of iterations to reach the steady state solution. The pressure and velocity contour for filter without any outlet impedance and filter with additional impedance have been shown in Fig.26, respectively. The steady solutions have been exported to data file and been stored in local memory. Similar to laminar flow analysis, the transient analysis has been carried out with above mentioned modification. As the flow is at high Mach number, two different microphone spacing have been adapted to estimate the acoustic TL. The microphone spacing of 0.1m and 0.2m have been taken to estimate acoustic TL in the ranges 170-1355 Hz and 85-678 Hz, respectively. The first mode plane wave cutoff frequency

26

for the analyzed filter is 1004Hz. First, for the microphone spacing 0.1m, the simulated four-pole signals for acoustic filter without outlet impedance have been shown in Fig.27 and for acoustic filter with outlet impedance have been shown in Fig.28. The estimated acoustic TL following four-pole method has been shown in Fig.29. From Fig.29(a), it is worth noticing that the estimated acoustic TL agrees to experimental result for a frequency range 170-1400 Hz. The estimated acoustic TL using microphone spacing 0.2m has been shown in Fig.29(b) which agrees to experimental result adequately in 50-750 Hz range. 7. Conclusion From above analyses, finite element based frequency domain analysis following three-pole measurement method is recommended to estimate acoustic TL without flow. Only the acoustic cavity needs to be modeled and the boundary layer meshing is not required. The simulation of anechoic termination can be attained by applying characteristic acoustic impedance of air on outlet surface. From numerical complexity stand point, the estimation of acoustic TL of a muffler with presence of mean flow, the pressure based solver and four-pole with two-load measurement method is recommended. The velocity inlet and pressure outlet are recommended boundary conditions for inlet and outlet, respectively. An additional filter attached to outlet is suitable in simulating a load situation. In presence of mean flow, the use of boundary layer meshing and the initialization of transient analysis using a steady state mean flow solution is advised. In transient analysis, after successive simulation of velocity impulse, the velocity inlet is recommended to assign non-reflecting boundary. An optimal numerical stability in simulating acoustic plane wave is achieved with 0.05th of the recommended time step or much lower. The smaller time steps increase the numerical stability proportionally. In order to reduce computational overhead, without presence of mean flow, the mapped meshing is advised. Summarizing, the proposed method can be reckoned in 27

designing industrial acoustic filter of any arbitrary shape. Acknowledgement One of the authors, Dibya Prakash Jena, acknowledges the generous funding received from the Ministry of Human Resource Development (MHRD), Government of India, for carrying out this work at Indian Institute of Technology, Bhubaneswar.

28

References [1] M. L. Munjal, Acoustics of Ducts and Mufflers, 2nd Edition, John Wiley and Sons, 2014. [2] M. L. Munjal, Recent advances in muffler acoustics, International Journal of Acoustics and Vibration 18 (2013) 71–85. [3] E. Dokumaci, Effect of sheared grazing mean flow on acoustic transmission in perforated pipe mufflers, Journal of Sound and Vibration 283 (2005) 645 – 663. [4] N. K. Vijayasree, M. L. Munjal, On an Integrated Transfer Matrix method for multiply connected mufflers, Journal of Sound and Vibration 331 (2012) 1926–1938. [5] S. Bilawchuk, K. Fyfe, Comparison and implementation of the various numerical methods used for calculating transmission loss in silencer systems, Applied Acoustics 64 (2003) 903 – 916. [6] J. Zhenlin, M. Qiang, Z. Zhihua, Application of the boundary element method to predicting acoustic performance of expansion chamber mufflers with mean flow, Journal of Sound and Vibration 173 (1994) 57 – 71. [7] C. I. Papadopoulos, Development of an optimized, standard-compliant procedure to calculate sound transmission loss: numerical measurements, Applied Acoustics 64 (2003) 1069 – 1085. [8] D. Siano, Three-dimensional/one-dimensional numerical correlation study of a three-pass perforated tube, Simulation Modelling Practice and Theory 19 (2011) 1143 – 1153.

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[9] A. Broatch, J. Serrano, F. Arnau, D. Moya, Time-domain computation of muffler frequency response: Comparison of different numerical schemes, Journal of Sound and Vibration 305 (2007) 333–347. [10] Y.-B. Park, H.-D. Ju, S.-B. Lee, Transmission loss estimation of threedimensional silencers by system graph approach using multi-domain BEM, Journal of Sound and Vibration 328 (2009) 575–585. [11] ASTM-E2611, Standard test method for measurement of normal incidence sound transmission of acoustical materials based on the transfer matrix method, ASTM International standard, 2009. [12] R. Kirby, A comparison between analytic and numerical methods for modelling automotive dissipative silencers with mean flow, Journal of Sound and Vibration 325 (2009) 565 – 582. [13] L. Xiang, S. Zuo, X. Wu, J. Liu, Study of multi-chamber microperforated muffler with adjustable transmission loss, Applied Acoustics 122 (2017) 35 – 40. [14] X. Shi, C.-M. Mak, Sound attenuation of a periodic array of microperforated tube mufflers, Applied Acoustics 115 (2017) 15 – 22. [15] Y.-C. C. Min-Chie Chiu, An assessment of high-order-mode analysis and shape optimization of expansion chamber mufflers, Archives of Acoustics 39 (2014) 489499. [16] C.-I. J. Young, M. J. Crocker, Prediction of transmission loss in mufflers by the finite element method, The Journal of the Acoustical Society of America (1975) 144–148. [17] O. Z. Mehdizadeh, M. Paraschivoiu, A three-dimensional finite element approach for predicting the transmission loss in mufflers and silencers with no mean flow, Applied Acoustics 66 (2005) 902 – 918. 30

[18] Z. Tao, A. Seybert, A review of current techniques for measuring muffler transmission loss, SAE Technical Paper: 2003-01-1653. [19] ANSYS, ANSYS/Fluent 14.5 user manual, 2013. [20] G. Montenegro, A. Onorati, A. D. Torre, The prediction of silencer acoustical performances by 1d, 1d-3d and quasi-3d non-linear approaches, Computers & Fluids 71 (2013) 208 – 223. [21] P. Wesseling, Principles of computational fluid dynamics, 1st Edition, Springer-Verlag Berlin Heidelberg, 2001. [22] G. Batchelor, An introduction to fluid dynamics, 1st Edition, Cambridge Univ.Press, 1967. [23] S. Patankar, Numerical heat transfer and fluid fLow, 1st Edition, Hemisphere Publishing Corporation, 1980. [24] K. Thompson, Time dependent boundary conditions for hyperbolic systems ii, Journal of Computational Physics 89 (1990) 439 – 461. [25] F. Piscaglia, A. Montorfano, A. Onorati, Development of a nonreflecting boundary condition for multidimensional nonlinear duct acoustic computation, Journal of Sound and Vibration 332 (2013) 922 – 935. [26] L. Selle, F. Nicoud, T. Poinsot, Actual impedance of nonreflecting boundary conditions: Implications for computation of resonators, AIAA Journal 42 (2004) 958–964. [27] A. Torregrosa, P. Fajardo, A. Gil, R. Navarro, Development of nonreflecting boundary condition for application in 3d computtational fluid dynamics code, Enginnering Appllications of Computtational Fluid Mechanics 6 (2012) 447–460.

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[28] M.-H. Hu, J.-S. Wu, Y.-S. Chen, Development of a parallelized 2d/2daxisymmetric navier-stokes equation solver for all-speed gas flows, Computers & Fluids 45 (2011) 241 – 248. [29] Z. Chen, A. Przekwas, A coupled pressure-based computational method for incompressible/compressible flows, Journal of Computational Physics 229 (2010) 9150 – 9165. [30] W. Polifke, C. Wall, P. Moin, Partially reflecting and non-reflecting boundary conditions for simulation of compressible viscous flow, Journal of Computational Physics 213 (2006) 437 – 449. [31] S. Kumar, Linear acoustic modelling and testing of exhaust mufflers, in: Master of Science Thesis Stockholm, Sweden, 2005. [32] D. Wilcox, Turbulence modeling for CFD, 1st Edition, DCW Industries, Inc. La Canada, California, 1998.

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Figure Captions Fig.1: The schematic of acoustic transmission loss measurement; (a) threepole and (b) four-pole with two load method. Fig.2: (a) 3D model of simple expansion chamber, (b) corresponding mesh using fluid-30 element, (c) error in applying boundary load and (d) recommended way in applying boundary load. Fig.3: The PSD of pressure signals (absolute) for three-pole measurement; (a) position m1 , (b) position m2 and (c) position m3 . Fig.4: Estimated acoustic transmission loss using finite element method (frequency domain analysis) and experimental result. Fig.5: Progressive plane wave simulation with anechoic termination on outlet; (a) 2D-axisyymetric duct, (b) corresponding mapped mesh and (c) boundary layer mesh. Fig.6: Pressure distribution for a fluctuation of 10 Pa at 500 Hz on pressure inlet for different time steps; (a) 1e-5 second (mapped mesh), (b) 1e-6 second (mapped mesh), (c) 1e-5 second (boundary layer mesh) and (b) 1e-6 second (boundary layer mesh). Fig.7: Pressure variation on axis for a fluctuation of 10 Pa at 500 Hz on pressure inlet for different time steps; (a) mapped mesh and (b) boundary layer mesh. Fig.8: (a) 2D-axisymmetric model of simple expansion chamber, (b) corresponding mapped mesh. Fig.9: Simulated pressure signals with non-reflecting pressure outlet boundary condition using density based solver; (a) position m1 , (b) position m2 , (c) position m3 and (d) position m4 . Fig.10: Estimated acoustic transmission loss using three-pole measurement method (density based solver) and experimental result. Fig.11: Simulated pressure signals with an open-end boundary condition using density based solver; (a) position m1 , (b) position m2 , (c) position m3 and (d) position m4 . 33

Fig.12: Simulated pressure signals with closed-end boundary condition using density based solver; (a) position m1 , (b) position m2 , (c) position m3 and (d) position m4 . Fig.13: Estimated acoustic transmission loss using four-pole with two load measurement method using density based solver and experimental result. Fig.14: Simulated axial particle velocity signals using density based solver; (a) position m1 with NRBC outlet, (b) position m3 with NRBC outlet, (c) position m1 with open outlet, (d) position m3 with open outlet, (e) position m1 with closed outlet, and (f) position m3 with closed outlet. Fig.15: Estimated acoustic transmission loss using pressure and particle velocity at two microphone positions following four-pole−two load measurement method using density based solver and experimental result. Fig.16: Simulated pressure signals with non-reflecting pressure outlet boundary condition using pressure based solver; (a) position m1 , (b) position m2 , (c) position m3 and (d) position m4 . Fig.17: Estimated acoustic transmission loss using three-pole measurement method (pressure based solver) and experimental result. Fig.18: Estimated acoustic transmission loss using four-pole−two-load method measurement technique using pressure based solver and experimental result. Fig.19: Simulated pressure signals with additional filter on outlet boundary using pressure based solver; (a) additional filter (load-2), (b) position m1 , (c) position m2 , (d) position m3 and (e) position m4 . Fig.20: Estimated acoustic transmission loss using four-pole-two-load method using identical muffler as a load condition. Fig.21: Boundary layer mesh of simple expansion chamber. Fig.22: Steady state solution (laminar flow); (a) velocity profile (without impedance), (b) total pressure profile along the axis (without impedance); (c) velocity profile (with impedance), and (d) total pressure profile along the axis (with impedance). Fig.23: Simulated pressure signals with presence on mean flow and open pres-

34

sure outlet boundary condition using pressure based solver (Mach number 2.92e-5); (a) position m1 , (b) position m2 , (c) position m3 and (d) position m4 . Fig.24: Simulated pressure signals with presence on mean flow and additional filter on outlet boundary using pressure based solver (Mach number 2.92e-5); (a) position m1 , (b) position m2 , (c) position m3 and (d) position m4 . Fig.25: Estimated acoustic transmission loss using four-pole with two load measurement method. Fig.26: Steady state solution (turbulent flow); (a) velocity profile (without impedance), (b) total pressure profile along the axis (without impedance); (c) velocity profile (with impedance), and (d) total pressure profile along the axis (with impedance). Fig.27: Simulated pressure signals with presence on mean flow and open pressure outlet boundary condition using pressure based solver (Mach number 0.06, and microphone spacing 0.1m); (a) position m1 , (b) position m2 , (c) position m3 and (d) position m4 . Fig.28: Simulated pressure signals with presence on mean flow and additional filter on outlet boundary using pressure based solver (Mach number 0.06, and microphone spacing 0.1m); (a) position m1 , (b) position m2 , (c) position m3 and (d) position m4 . Fig.29: Estimated acoustic transmission loss using four-pole with two load measurement method using (a) microphone spacing 0.1m., and (b) microphone spacing 0.2m.

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